JOURNAL OF CHEMICAL PHYSICS
VOLUME 115, NUMBER 15
15 OCTOBER 2001
Analysis of the linear response function along the adiabatic connection
from the Kohn–Sham to the correlated system
Andreas Savin,a) François Colonna, and Marcel Allavena
Laboratoire de Chimie Théorique (CNRS), Université Pierre et Marie Curie 4, Place Jussieu,
F-75252 Paris, France
共Received 25 June 2001; accepted 1 August 2001兲
Careful calculations are performed to obtain the radial density–density response function for the He
and the Be series. This is also done along the adiabatic connection of the density functional theory
共as the system evolves from the real, physical system to the Kohn–Sham one兲. In this process the
electron density is kept constant, while the strength of the interaction between electrons changes.
The response functions are analyzed in terms of their eigenvalues and eigenfunctions. The latter
change only little along this process. The absolute value of the eigenvalues is in general reduced by
the interaction: A screening effect is present. For the near-degenerate systems, we notice that the
opposite effect can appear 共antiscreening兲. © 2001 American Institute of Physics.
关DOI: 10.1063/1.1405011兴
I. INTRODUCTION
physical and Kohn–Sham systems; it is the way most approximations are understood. Although it is not necessary
共cf. Refs. 3– 6兲, it is usually preferred to keep the density
constant along the adiabatic connection.7–11 In order to
achieve it, the external potential changes continuously from
v ext to v KS : v ext→ v ␭ 共v ␭⫽1 ⫽ v ext , v ␭⫽0 ⫽ v KS兲. We will also
show in the following some results for the linear response
function along the adiabatic connection ␹ (␭)⫽ ␹ (r,r⬘ ;␭):
The Hohenberg–Kohn theorem1 establishes the mapping
between the external potential of a given system, v ext , and
the ground state density n. As a chemical process implies a
change in v ext it seems natural in this context, as a first step,
to study the relationship between small changes in v ext ,
␦ v ext , and the corresponding changes in n, ␦ n. The connection between these quantities is given by the linear response
function ␹ (r,r⬘ ):
␦ n 共 r兲 ⫽
冕␹
共 r,r⬘ 兲 ␦ v ext共 r⬘ 兲 d 3 r⬘ ,
␦ n⫽ ␹ 共 ␭ 兲 쐓 ␦ v ␭ .
Usually, approximations are made only for the
exchange-correlation part of the energy, which corresponds
to approximating the exchange-correlation potential, v xc :
共1兲
where ␹ (r,r⬘ ) is defined perturbationally. We use a compact
notation by introducing the star symbol 共쐓兲:
v KS⫽ v ext⫹ v h ⫹ v xc ,
␦ n⫽ ␹ 쐓 ␦ v ext .
where v h ⫽ 兰 (n/r 12)d 3 r 2 . Now using the fact that the same
density change is obtained by ␦ v KS and ␦ v ext , and after
introducing the inverse response functions 共see, e.g., Ref.
⫺1
,
12兲, ␹ ⫺1 , ␹ KS
The aim of this study is to provide more information
about ␹ by performing reasonably accurate calculations for
small systems: the He series and the Be series.
The second important aspect of density functional theory
共DFT兲 is the Kohn–Sham approach:2 The vast majority of
practical calculations are performed by considering the fictitious system of noninteracting electrons, yielding the same
density as the 共physical兲 system of interest. As the
Hohenberg–Kohn theorem is valid for any type of interaction between particles, there also is a mapping between n and
the potential in the Kohn–Sham system, v KS . It is thus interesting to study the corresponding Kohn–Sham response
function ␹ KS :
␦ v ext⫽ ␹ ⫺1 쐓 ␦ n,
one obtains
⫺1
␹ KS
⫽ ␹ ⫺1 ⫹
where f xc⫽ ␦ v xc / ␦ n. Thus the problem of finding ␹ or
( ␹ ⫺1 ) is transferred, in the Kohn-Sham approach, to that of
finding f xc .
In the simplest approach, the random phase approximation 共RPA兲, f xc is set to zero:
␦ n⫽ ␹ KS쐓 ␦ v KS .
⫺1
⫺1
␹ RPA
⫽ ␹ KS
⫺
It is usually assumed that it is possible to reach the Kohn–
Sham system by switching off the electron–electron interaction 共1/r 12→␭/r 12 ; 0⭐␭⭐1兲. This adiabatic connection
plays a central role in DFT, being the bridge between the
1
.
r 12
Please note that ␹ RPA contains a self-interaction error, as for
any one-electron system, ␹ is equal to ␹ KS and not to ␹ RPA .
This error is also present in the popular local density approximation, where v xc(r) is a function of n(r), and
a兲
Electronic mail: [email protected]
0021-9606/2001/115(15)/6827/7/$18.00
1
⫹ f xc ,
r 12
6827
© 2001 American Institute of Physics
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6828
J. Chem. Phys., Vol. 115, No. 15, 15 October 2001
Savin, Colonna, and Allavena
f xc(r,r⬘ )⫽ ␦ v xc(r)/ ␦ n(r⬘ ) thus has the form f xc(n(r)) ␦ (r
⫺r⬘ ). Of course, this is also wrong, as for a one-electron
system f xc(r,r⬘ )⫽⫺1/兩 r⫺r⬘ 兩 .
The connection between ␦ v ext and ␦ v KS is given by
where ⑀ ⫺1 is the inverse dielectric function, which can be
written as
⫺1
⑀ ⫺1 ⫽ ␹ KS
쐓␹.
共2兲
Physically, ⑀ ⫺1 reflects the effect of the interaction between electrons, and one might expect that this produces a
screening of ␦ v ext and thus a reduction of ␹ with respect to
␹ KS ( ⑀ ⫺1 ⬍1). One of the objectives of this work is to check
whether this is true in the systems considered.
With ␹共␭兲 one has to deal with functions of six variables.
As we are interested in ␦ v in atoms, we develop these functions in terms of spherical harmonics,
兺
兺
l,m
l ⬘ ,m ⬘
␹ 共 r,l,m;r ⬘ ,l ⬘ ,m ⬘ ;␭ 兲
⫻Y lm 共 ⍀ 兲 Y l ⬘ m ⬘ 共 ⍀ ⬘ 兲 .
As we consider only spherical densities ␦ n(r)⫽ ␦ n(r),
and furthermore only spherical potentials ␦ v (r)⫽ ␦ v (r).
Thus,
␦n共 r 兲⫽
冕␹
共 r,l⫽0,m⫽0;r ⬘ ,l ⬘ ⫽0,m ⬘ ⫽0 兲
⫻ ␦ v ext共 r ⬘ 兲 r ⬘ 2 dr ⬘ .
To simplify notation, we will drop in what follows l⫽l ⬘
⫽m⫽m ⬘ ⫽0, and write
1
␹ r共 r,r ⬘ ;␭ 兲 ⫽
␹ 共 r,l⫽0,m⫽0;r ⬘ ,l ⬘ ⫽0,m ⬘ ⫽0;␭ 兲 .
4␲
共3兲
In order to manipulate the information we will consider the
eigenvector decomposition, see, e.g., Ref. 13:
␹ r 共 r,r ⬘ ;gl 兲 ⫽
兺i ␬ i共 ␭ 兲 ␺ i共 r;␭ 兲 ␺ i共 r ⬘ ;␭ 兲 ,
where the ␺ i are orthonormal:
4␲
冕
r 2 ␺ i 共 r;␭ 兲 ␺ j 共 r;␭ 兲 dr⫽ ␦ i j .
One advantage of this decomposition is that we can get the
best approximation to ␹ r 共in a least-squares sense兲 by cutting
off the above-mentioned expansion by selection of the terms
corresponding to the most important eigenvalues. Notice that
a change of the potential by a constant does not modify the
density, implying 兰 ␹ ⫽0; this in turn means that
4␲
冕
The response functions are derived from perturbation
theory, by using
␦ n 共 r兲 ⫽2 具 ⌿ 共 0 兲 兩 ␳ˆ 共 r兲 兩 ⌿ 共 1 兲 典 ,
␦ v KS⫽ ⑀ ⫺1 쐓 ␦ v ext ,
␹ 共 r,r⬘ ;␭ 兲 ⫽
II. METHOD
r 2 ␺ i 共 r 兲 dr⫽0
so that the ␺ i (r) necessarily have nodes.
where ⌿ (0) is the unperturbed ground state wave function,
⌿ (1) the first-order correction to it, due to the perturbing
potential ␦ v (r⬘ ), and ␳ˆ (r) the density operator. To obtain
␹ r (r,r ⬘ ) we use Eqs. 共1兲 and 共3兲:
␦ n 共 r兲 ⫽ ␦ n 共 r 兲 ⫽
冕␹
r 共 r,r̃ 兲 ␦ v共 r̃ 兲 4 ␲ r̃
2
dr̃.
共4兲
Thus, ␹ r (r,r ⬘ ) is ␦ n(r) obtained with ␦ v (r̃)⫽ ␦ (r̃
⫺r ⬘ )/4␲ r̃ 2 . In some cases, such as the hydrogen atom, ⌿ (1)
and thus ␹ r can be obtained analytically. In many cases, like
the noninteracting systems, it is common to use the sumover-states formula to obtain ⌿ (1) and thus ␹ r . With many
programs, it is also convenient to use a small finite perturbing potential and analyze the resulting density.
For the noninteracting closed-shell system, it is easy to
use the expression of ␹ KS in terms of Kohn–Sham orbitals,
␸ i , and eigenvalues ⑀ i 共in a finite basis兲:
occ unocc
兺i 兺a
␹ KS共 r,r⬘ 兲 ⫽4
␸ i 共 r兲 ␸ a 共 r兲 ␸ i 共 r⬘ 兲 ␸ a 共 r⬘ 兲
.
⑀ i⫺ ⑀ a
共5兲
For interacting systems we generate ␹ r (␭) in the basis of
the ␺ i (r;␭⫽0). To obtain it, we made a configuration interaction calculation after changing the external potential by
␦ v k (r)⫽10⫺3 ␺ k (r,␭⫽0), and decompose the resulting
density changes onto the same basis ␺ k (r,␭⫽0); from
␹ r 共 r,r ⬘ ;␭ 兲 ⫽
兺i j ␹ i j 共 ␭ 兲 ␺ i共 r;␭⫽0 兲 ␺ j 共 r ⬘ ;␭⫽0 兲 ,
␦ n 共 r;␭ 兲 ⫽ 兺
k
i
共6兲
␦ n ki 共 ␭ 兲 ␺ i 共 r;␭⫽0 兲 ,
we obtain that the change in ␦ n ki (␭)⫽ 兰 r 2 ␦ n k (r;␭) ␺ i (r;␭
⫽0)dr induced by ␦ v k will be, to linear order, 10⫺3 ␹ ik (␭).
Some technical details are given in Appendices A and B.
We now check some limits of the procedures just described. A first one is given by the finite size of the basis set.
One possibility to check it, is to compare the eigenvalues
␬ i (␭⫽0), for a given basis set, with those obtained from a
larger basis set, or those obtained on a dense numerical grid,
by using the exact response function of the hydrogen atom
共cf. Appendix C兲. We show in Fig. 1 the decimal logarithm
of the first 35 eigenvalues ␬ i (␭⫽0) of ␹ KS of the H atom:
from calculations on a grid with 256 points and on two
Gaussian basis sets. Of course, an increase of the size of the
basis set produces a larger domain of validity of the ␬ i ’s.
Figure 1 shows a significant loss of quality of the smaller
eigenvalues 共notice the logarithmic scale of the plot兲. It is
clear that only a limited number of nodes can be represented
with finite basis sets. It is thus not surprising that not all
eigenvalues are pertinent. Notice also that ␹ r has a number
of eigenvalues larger than the number of basis functions. In
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J. Chem. Phys., Vol. 115, No. 15, 15 October 2001
Adiabatic connection
6829
TABLE I. First seven eigenvalues of the response function in the He series.
For each system, ␬ i (␭⫽0), ␬ i (␭⫽1), and their ratio ˜⑀ ⫺1
i ⫽ ␬ i (␭
⫽1)/ ␬ i (␭⫽0) are shown 共first, second, and third lines, respectively兲.
FIG. 1. Decimal logarithm of the 35 first eigenvalues of the hydrogen atom
response function. The results obtained on a dense numerical grid are shown
as dots, while calculations with 31 and 21 Gaussian s functions are shown as
triangles and squares, respectively.
fact, the problem of not describing the small eigenvalues is
related to that of linear dependencies in the products of basis
functions 共see e.g., Refs. 14 –16兲.
The second problem is the finite perturbation approach
which was used to obtain ␹ (␭⫽0). For ␹ (␭⫽0) the exact
␦ n ki are equal to zero for i⫽k. Due to the finite perturbation,
these ␦ n ki are, however, of the order of ( ␦ v k ) 2 . For example,
for the hydrogen atom, using ␦ v 1 (r)⫽10⫺3 ␺ 1 (r), the ␦ n 11
are of the order of 10⫺4 while the ␦ n 1i (i⫽1) are of the order
of 10⫺8 or smaller.
III. RESULTS
Typical plots of r 2 ␹ (r,r ⬘ )r ⬘ 2 are shown in Fig. 2, taking
as examples the hydrogen, helium, and beryllium atoms. The
plot for the H atom shows that there are essentially two regions in the atom 共the inner sphere, and the outer shell兲: A
repulsive potential change ␦ v in one of the regions produces
a displacement of the radial density from it to the other region. Of course, the opposite effect appears for an attractive
␦ v . The plot for the He atom is similar 共cf. Fig. 2兲. For the
Be atom, however, a duplication appears, which we attribute
to the shell structure. Such plots can be made for all ␭. As the
figures look similar, we show only ␭⫽0.
We will continue now with the analysis of the eigenvalues, ␬ i (␭), and eigenfunctions, ␺ i (␭), of the ␹ r (␭). The
eigenvalues are given for ␭⫽0 and ␭⫽1 in Tables I and II.
Ordering the eigenfunctions with increasing eigenvalues 共the
i⫽1
i⫽2
i⫽3
i⫽4
i⫽5
i⫽6
i⫽7
He
⫺0.51
⫺0.40
0.79
⫺0.09
⫺0.08
0.97
⫺0.04
⫺0.04
1.00
⫺0.02
⫺0.02
1.00
⫺0.01
⫺0.01
1.00
⫺0.01
⫺0.01
1.00
⫺0.01
⫺0.01
1.00
Li⫹
⫺0.74
⫺0.65
0.87
⫺0.13
⫺0.13
0.98
⫺0.06
⫺0.06
1.00
⫺0.04
⫺0.04
1.00
⫺0.02
⫺0.02
1.00
⫺0.02
⫺0.02
1.00
⫺0.01
⫺0.01
1.00
Be⫹2
⫺0.99
⫺0.89
0.90
⫺0.18
⫺0.18
0.99
⫺0.08
⫺0.08
1.00
⫺0.05
⫺0.05
1.00
⫺0.03
⫺0.03
1.00
⫺0.02
⫺0.02
1.00
⫺0.01
⫺0.01
1.00
B⫹3
⫺1.23
⫺1.14
0.92
⫺0.23
⫺0.23
0.99
⫺0.11
⫺0.10
1.00
⫺0.06
⫺0.06
1.00
⫺0.04
⫺0.04
1.00
⫺0.03
⫺0.03
1.00
⫺0.02
⫺0.02
1.00
C⫹4
⫺1.48
⫺1.39
0.94
⫺0.28
⫺0.28
0.99
⫺0.13
⫺0.13
1.00
⫺0.07
⫺0.07
1.00
⫺0.05
⫺0.05
1.00
⫺0.03
⫺0.03
1.00
⫺0.02
⫺0.02
1.00
N⫹5
⫺1.72
⫺1.63
0.95
⫺0.33
⫺0.32
0.99
⫺0.15
⫺0.15
1.00
⫺0.09
⫺0.09
1.00
⫺0.06
⫺0.06
1.00
⫺0.04
⫺0.04
1.00
⫺0.03
⫺0.03
1.00
O⫹6
⫺1.97
⫺1.88
0.95
⫺0.37
⫺0.37
0.99
⫺0.17
⫺0.17
1.00
⫺0.10
⫺0.10
1.00
⫺0.06
⫺0.06
1.00
⫺0.05
⫺0.05
1.00
⫺0.03
⫺0.03
1.00
F⫹7
⫺2.22
⫺2.13
0.96
⫺0.42
⫺0.42
0.99
⫺0.19
⫺0.19
1.00
⫺0.11
⫺0.11
1.00
⫺0.07
⫺0.07
1.00
⫺0.05
⫺0.05
1.00
⫺0.04
⫺0.04
1.00
Ne⫹8
⫺2.46
⫺2.37
0.96
⫺0.47
⫺0.47
0.99
⫺0.22
⫺0.21
1.00
⫺0.12
⫺0.12
1.00
⫺0.08
⫺0.08
1.00
⫺0.06
⫺0.06
1.00
⫺0.04
⫺0.04
1.00
eigenvalues are negative, see, e.g., Ref. 17兲 we observe that
in the Be series the first four eigenfunctions are localized in
the core region, while the fifth is localized in the valence
region 共cf. Fig. 3 for Be at ␭⫽0兲. It turns out that the eigenfunctions change only little with ␭; we notice a significant
change in the eigenvalues, especially for the first one in each
shell. We show in Figs. 4 and 5 the ratio ␬ i (␭)/ ␬ i (␭⫽0) for
the helium and the beryllium atoms. We can also decompose
FIG. 2. Isocontour plot of r 2 r ⬘ 2 ␹ r (r,r ⬘ ;␭⫽0) for the H, He, and Be atoms, from left to right, respectively. The zero contour has been emphasized;
␹ r (r,r ⬘ ⫽r;␭⫽0)⬍0.
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6830
J. Chem. Phys., Vol. 115, No. 15, 15 October 2001
Savin, Colonna, and Allavena
TABLE II. First seven eigenvalues of the response function in the Be series.
For each system, ␬ i (␭⫽0), ␬ i (␭⫽1), and their ratio ˜⑀ i⫺1 ⫽ ␬ i (␭
⫽1)/ ␬ i (␭⫽0) are shown 共first, second, and third lines, respectively兲.
i⫽1
i⫽2
i⫽3
i⫽4
i⫽5
i⫽6
i⫽7
Be
⫺1.01
⫺0.90
0.90
⫺0.18
⫺0.18
0.98
⫺0.08
⫺0.08
1.00
⫺0.05
⫺0.05
1.00
⫺0.04
⫺0.04
0.88
⫺0.03
⫺0.03
1.00
⫺0.02
⫺0.02
1.00
B⫹1
⫺1.26
⫺1.16
0.92
⫺0.23
⫺0.23
0.99
⫺0.11
⫺0.11
1.00
⫺0.06
⫺0.06
1.00
⫺0.05
⫺0.05
0.95
⫺0.04
⫺0.04
1.00
⫺0.03
⫺0.03
1.00
C⫹2
⫺1.52
⫺1.42
0.93
⫺0.28
⫺0.28
0.99
⫺0.13
⫺0.13
1.00
⫺0.07
⫺0.07
1.00
⫺0.06
⫺0.06
0.98
⫺0.05
⫺0.05
1.00
⫺0.03
⫺0.03
1.00
N⫹3
⫺1.78
⫺1.68
0.95
⫺0.33
⫺0.33
0.99
⫺0.15
⫺0.15
1.00
⫺0.09
⫺0.09
1.00
⫺0.07
⫺0.07
1.01
⫺0.06
⫺0.06
1.00
⫺0.04
⫺0.04
1.00
O⫹4
⫺2.03
⫺1.94
0.96
⫺0.38
⫺0.38
1.00
⫺0.17
⫺0.17
1.00
⫺0.10
⫺0.10
1.00
⫺0.08
⫺0.09
1.04
⫺0.07
⫺0.07
1.00
⫺0.05
⫺0.05
1.01
F⫹5
⫺2.29
⫺2.21
0.96
⫺0.43
⫺0.43
1.00
⫺0.20
⫺0.20
1.00
⫺0.11
⫺0.11
1.00
⫺0.09
⫺0.10
1.07
⫺0.07
⫺0.08
1.00
⫺0.05
⫺0.05
1.01
Ne⫹6
⫺2.55
⫺2.47
0.97
⫺0.48
⫺0.48
1.00
⫺0.22
⫺0.22
1.00
⫺0.13
⫺0.13
1.00
⫺0.11
⫺0.12
1.09
⫺0.08
⫺0.08
1.00
⫺0.06
⫺0.06
1.01
␹ r (␭) on the basis of the eigenfunctions of ␹ r (␭⫽0),
␺ i (␭⫽0), as in Eq. 共6兲. Within our accuracy, ␹ i j is a nearly
diagonal matrix, as can be seen in Fig. 6. This in turn means
that ⑀ ⫺1 关Eq. 共2兲兴 will also be nearly diagonal in this basis:
⑀ ⫺1 共 r,r ⬘ ;␭ 兲 ⫽ 兺 ⑀ ⫺1
i j 共 ␭ 兲 ␺ i 共 r;␭⫽0 兲 ␺ j 共 r ⬘ ;␭⫽0 兲
ij
⬇
兺i ˜⑀ ⫺1
i 共 ␭ 兲 ␺ i 共 r;␭⫽0 兲 ␺ j 共 r ⬘ ;␭⫽0 兲 ,
where ˜⑀ ⫺1
i (␭)⫽ ␬ i (␭)/ ␬ i (␭⫽0). For example, for the beryl-
FIG. 3. First and fifth eigenvectors of ␹ KS for the Be atom: r 2 ␺ 1 (r;␭⫽0)
共continuous line兲 and r 2 ␺ 5 (r;␭⫽0) 共dotted line兲.
FIG. 4. Dependence of the first few eigenvalues of the inverse dielectric
function, approximated by the ratio of the eigenvalues of ␹ ␭ :˜⑀ ⫺1
i
⫽ ␬ i (␭)/ ␬ i (␭⫽0) on the adiabatic coupling constant ␭. The lowest curve
corresponds to i⫽1; for the He atom.
lium atom, we have for i⫽5 共the most important valence
⫺1
eigenstate兲 ˜⑀ ⫺1
5 (␭⫽1)⫽0.884 within 0.5% of ⑀ 55 (␭⫽1)
⫽0.888.
We now turn to the question of screening. We will speak
about ‘‘screening’’ when ˜⑀ ⫺1
i ⬍1, and about ‘‘antiscreening’’
⫺1
⬎1.
Some
˜
⑀
were
already shown in Figs. 4 and
when ˜⑀ ⫺1
i
i
5. We see that in theses cases screening is present. We now
show in Fig. 7 the plot of ˜⑀ ⫺1 (␭) for Ne6⫹. It turns out that
˜⑀ ⫺1
5 significantly increases with ␭. Thus, for the valence shell
of Ne6⫹ antiscreening appears. This change appears for N3⫹
in the Be series, as can be seen in Fig. 8, where ˜⑀ ⫺1
5 (␭
⫽1) is shown as a function of nuclear charge of the ion, Z.
We could not detect, however, antiscreening effects at RPA
level. We thus attribute antiscreening to the increase in
␬ i (␭)/ ␬ i (␭⫽0) produced by f xc .
In order to understand the antiscreening in the Be series,
remember that two configurations 共⌽ s and ⌽ p , corresponding to 1s 2 2s 2 and 1s 2 2 p 2 兲 are both important. We should
FIG. 5. Dependence of the first few eigenvalues of the inverse dielectric
function, approximated by the ratio of the eigenvalues of ␹ ␭ :˜⑀ ⫺1
i
⫽ ␬ i (␭)/ ␬ i (␭⫽0) on the adiabatic coupling constant ␭. From the lowest
curve to the uppermost curve, we have i⫽5,1,2,3,4 for the Be atom.
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J. Chem. Phys., Vol. 115, No. 15, 15 October 2001
Adiabatic connection
6831
FIG. 6. Plots of the ␹ i␭⫽1
matrix elej
ments on the basis of the eigenvectors
of ␹ ␭⫽0 for He, Be, and Ne6⫹ 共lefthand side, middle, and right-hand
side兲.
thus consider, as a first approximation to the eigenstates 共see
also Ref. 18兲:
⌿ 共00 兲 ⫽cos ␣ ⌽ s ⫺sin ␣ ⌽ p ,
⌿ 共10 兲 ⫽sin ␣ ⌽ s ⫹cos ␣ ⌽ p .
To zeroth order, the response function is
␹
共0兲
共 r,r⬘ 兲 ⫽
具 ⌿ 共00 兲 兩 ␳ˆ 共 r兲 兩 ⌿ 共10 兲 典具 ⌿ 共10 兲 兩 ␳ˆ 共 r⬘ 兲 兩 ⌿ 共00 兲 典
⌬E 01
⫹
兺
具 ⌿ 共00 兲 兩 ␳ˆ 共 r兲 兩 ⌿ 共i 0 兲 典具 ⌿ 共i 0 兲 兩 ␳ˆ 共 r⬘ 兲 兩 ⌿ 共00 兲 典
i⬎1
⌬E 0i
, 共7兲
the unwhere ⌬E 0i are the energy denominators, and ⌿ (0)
i
perturbed eigenfunctions. The first term on the right-hand
side reflects the effect of near degeneracy, while the second
is general. As ⌽ s and ⌽ p differ by two orbitals, and ␳ˆ (r) is
a one-electron operator, 具 ⌽ s 兩 ␳ˆ (r) 兩 ⌽ p 典 ⫽0 and the first term
on the right-hand side of Eq. 共7兲 has a numerator:
cos2 ␣ sin2 ␣ 共 具 ⌽ p 兩 ␳ˆ 共 r兲 兩 ⌽ p 典 ⫺ 具 ⌽ s 兩 ␳ˆ 共 r兲 兩 ⌽ s 典 兲
⫻共 具 ⌽ p 兩 ␳ˆ 共 r⬘ 兲 兩 ⌽ p 典 ⫺ 具 ⌽ s 兩 ␳ˆ 共 r⬘ 兲 兩 ⌽ s 典 兲 .
共8兲
The second term has contributions of the type
cos2 ␣ 共 具 ⌽ s 兩 ␳ˆ 共 r兲 兩 ⌽ v 典 ⫺ 具 ⌽ v 兩 ␳ˆ 共 r兲 兩 ⌽ s 典 兲 ,
共9兲
where ⌽ v ⫽⌽ s or ⌽ p , or
FIG. 7. Dependence of the first few eigenvalues of the inverse dielectric
function, approximated by the ratio of the eigenvalues of ␹ ␭ :˜⑀ ⫺1
i
⫽ ␬ i (␭)/ ␬ i (␭⫽0) on the adiabatic coupling constant ␭. The lowest curve
corresponds to i⫽1. From the lowest curve to the uppermost curve, we have
i⫽1,2,3,4,5 for the Ne6⫹ ion.
sin2 ␣ 共 具 ⌽ p 兩 ␳ˆ 共 r兲 兩 ⌽ v 典 ⫺ 具 ⌽ v 兩 ␳ˆ 共 r兲 兩 ⌽ p 典 兲 .
⌽ v contains monoexcitations from ⌽ s or ⌽ p ;
( 具 ⌽ s 兩 ␳ˆ (r) 兩 ⌽ v 典 and 具 ⌽ p 兩 ␳ˆ (r) 兩 ⌽ v 典 cannot both be nonzero,
and we thus do not have terms like 具 ⌽ s 兩 ␳ˆ (r) 兩 ⌽ v 典
⫻具 ⌽ v 兩 ␳ˆ (r) 兩 ⌽ p 典 兲. Thus when ␣ ⫽0 only terms of the type
Eq. 共9兲 are left. This corresponds to the noninteracting case
for the system considered. When ␣ is nonzero these terms,
Eq. 共9兲, become smaller than for ␣ ⫽0, reducing the response
共screening has occurred兲. Notice, however, that an opposite
effect is also present 关cf. Eq. 共8兲兴: A contribution to ␹ arises,
which was not present at ␣ ⫽0. We attribute to this term,
related to near degeneracy, the appearance of antiscreening.
The increase of antiscreening with the nuclear charge Z is
essentially related to the change of the energy denominator
⌬E 01 with Z.
IV. CONCLUSION AND PERSPECTIVES
In this paper we showed the static linear density–density
response function of Kohn–Sham systems having the densities of the He atom, the Be atom, and their isoelectronic ions,
up to Z⫽10. Furthermore, we studied the change of the response functions with ␭, a constant multiplying the electron–
electron interaction operator, while keeping the density constant. It turned out that the eigenstates of the response
functions do not significantly change with ␭, while their eigenvalues do. There is a significant difference between the
He and the Be series, in the change of the eigenvalues with
FIG. 8. Z-dependent antiscreening effect of the fifth component of the inverse dielectric function, approximated by the ratio of the eigenvalues of
␹ ␭ :˜⑀ ⫺1
5 ⫽ ␬ 5 (␭⫽1)/ ␬ 5 (␭⫽0) for the Be series.
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6832
J. Chem. Phys., Vol. 115, No. 15, 15 October 2001
Savin, Colonna, and Allavena
TABLE III. Central exponents ␣ c 关Eq. 共A1兲兴 of the even tempered Gaussian
basis sets.
System
s
pdf
He
Li⫹
Be2⫹
B3⫹
C4⫹
N5⫹
O6⫹
F7⫹
Ne8⫹
40.0
85.5
350.0
475.0
663.0
990.0
1200.0
1612.0
2030.0
3.0
7.5
15.0
23.0
34.0
48.0
65.0
82.5
100.0
Be
B⫹
C2⫹
N3⫹
O4⫹
F5⫹
Ne6⫹
4.0
4.6
10.0
7.5
12.0
100.0
130.0
1.0
3.3
5.0
8.0
11.0
9.0
15.0
APPENDIX B: CALCULATION OF THE RESPONSE
FUNCTION ␹ KS FROM THE EIGENVECTORS OF THE
OVERLAP MATRIX PRODUCTS OF PRIMITIVE
GAUSSIANS
Given a set of Gaussian primitive functions g p (r), let
␲ m ⫽g p (r)g q (r), where p runs over the primitives appearing
in the occupied orbitals and q runs over those appearing in
the virtual orbitals; m is a composite index. To avoid problems due to linear dependencies between the ␲ m , we choose
to develop the ␹ KS function on the eigenvectors ␩ a (r) of the
␲
⫽ 具 ␲ a兩 ␲ b典 ,
overlap matrix of the ␲ m , S a,b
S ␲ ␩ a 共 r兲 ⫽⌬ a ␩ a 共 r兲 ,
where the ⌬ a are the eigenvalues. The ⌬ a are sorted and only
those eigenvectors corresponding to (⌬ a ⬎10⫺12) are used in
the following. The orthogonal ␩ are developed on the ␲
functions with
␩ a ⫽ 兺 A a,m ␲ m 共 r兲 .
m
␭. While electron–electron interaction in general screens the
effect of the perturbing potential, we observe the opposite
effect for higher Z in the Be series. This effect is connected
with the contribution coming from the variation of the
exchange-correlation potential with the density, f xc , and
gives a further indication about the difficulty to properly describe it 共see, e.g., Ref. 19兲.
It can be hoped that a better knowledge of ␹ will finally
help one to construct better approximations to f xc , and to the
exchange-correlation functionals, in a similar way as knowledge of exact Kohn–Sham potentials are useful 共see, e.g.,
Ref. 20兲.
ACKNOWLEDGMENTS
The authors would like to thank John F. Dobson 共Nathan
University, Brisbane, Australia兲, Walter Kohn 共University of
California at Santa Barbara兲, and E. Zaremba 共Queen’s University, Kingston, Canada兲 for stimulating discussions. We
are grateful to the IDRIS 共CNRS, Orsay, France兲 for the
allocation of computing time.
Uncontracted even tempered Gaussian basis sets, up to f
functions, were used in the calculations. For each angular
quantum number, M exponents ␣ n were produced by the rule
n⫽1,...,M .
␩
␹ KS,ab
⫽
冕冕␹
occ
⫽
vir
KS共 r,r⬘ 兲 ␩ a 共 r 兲 ␩ b 共 r⬘ 兲 d
3
rd 3 r⬘
4
兺i 兺j ⑀ i ⫺ ⑀ j I ␩i ja I ␩i jb ,
where the overlap integrals I ␩i ja are given by
I ␩i ja ⫽
冕␸
i 共 r兲 ␸ j 共 r兲 ␩ a 共 r兲 d
3
r.
Diagonalization of ␹ ␩ in the ␩ a basis produces the orthonormal eigenvectors C a,i and the eigenvalues ␬ i . They, in
turn, give the development of the ␺ i (r) on the primitive
Gaussian pairs ␲ m (r):
␺ i 共 r兲 ⫽ 兺 C i,a ␩ a 共 r兲 ⫽ 兺 C i,a 兺 A a,m ␲ m 共 r兲 .
a
a
m
As for high i, the numerical values of the ␭ i may lose significance 共becoming 10⫺16 or lower or even positive兲, only
the first few largest in absolute value (⬍10⫺6 ) are kept 共19
for He series and 18 for Be series兲.
APPENDIX C: EXACT RESPONSE FUNCTION ␹ R OF
THE HYDROGEN ATOM
APPENDIX A: BASIS SETS
␣ n ⫽ ␣ c ␦ 共 2n⫺M ⫺1 兲 /2,
In the ␩ basis, using Eq. 共5兲, the matrix representation of
␹ KS(r,r⬘ ), ␹ ␩ , is given by
共A1兲
␦ ⫽2.1 for all systems. M has been chosen to be 21 for all the
s functions, 7 for the p functions of the helium series, 9 for
the p functions of the beryllium series, and 5 and 3 for the d
and f functions, respectively. The central exponents ␣ c are
given in Table III. For H 共Fig. 1兲 21 共with ␦ ⫽2.1兲 and 31
共with ␦ ⫽1.7兲 s-type functions, centered at ␣ c ⫽10 were
used.
In order to facilitate the reading of the present paper we
give in the following a derivation of ␹ r (r,r ⬘ ) for the hydrogen atom, although it is given, at least implicitly, in the literature 共see, e.g., Ref. 21兲.
In the case of noninteracting spherically symmetric systems it is customary to replace the radial functions R(r) by
u(r)⫽rR(r), in order to write the Schrödinger equation as
that of a particle in one dimension:
冋
⫺
册
1 d2
l 共 l⫹1 兲
⫹ v共 r 兲 u 共 r 兲 ⫽Eu 共 r 兲 ,
2⫹
2 dr
2r 2
where l is the angular momentum quantum number.
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J. Chem. Phys., Vol. 115, No. 15, 15 October 2001
Adiabatic connection
When the potential v (r) is decomposed into two parts,
v (r)⫽ v (0) (r)⫹ v (1) (r), and the solution for v (0) (r) is supposed to be known 共u (0) and E (0) 兲, the first-order equation is
written as
冋
This integral can be obtained analytically, e.g., with
24
MATHEMATICA. It is:
f 共 r,r ⬘ 兲 ⫽
册
1 d2
l 共 l⫹1 兲
⫺
⫹
⫹ v共 0 兲 ⫺E 共 0 兲 u 共 1 兲 ⫹ 关v 共 1 兲 ⫺E 共 1 兲 兴 u 共 0 兲
2 dr 2
2r 2
冋
册
df
d
(u 共 0 兲 ) 2
⫽2 共 v 共 1 兲 ⫺E 共 1 兲 兲共 u 共 0 兲 兲 2 .
dr
dr
冕
⬁
r
关v 共 1 兲 共 r̃ 兲 ⫺E 共 1 兲 兴共 u 共 0 兲 共 r̃ 兲兲 2 dr̃,
冕
dr
2
共 u 共 0 兲 共 r 兲兲 2
冕
⬁
r
E
⫽
冕冉
⫽
1
共 R 共 0 兲 共 r 0 兲兲 2
4␲
R
共r兲
1
冑4 ␲
冊
2
␦ 共 r̃⫺r 0 兲
4 ␲ r̃ 2 dr̃,
4 ␲ r̃ 2
where R (0) (r)⫽2e ⫺r , the radial function of the H atom. Inserting this result into Eq. 共C1兲 one gets
f 共 r; v 共 1 兲 共 r 兲 ⫽ ␦ 共 r̃⫺r ⬘ 兲 /4␲ r̃ 2 兲 ⬅ f 共 r;r ⬘ 兲 .
Thus
f 共 r,r ⬘ 兲 ⫽⫺
⫺
f 共 r,r ⬘ 兲 ⫽
冕
dr
共0兲
冊
共 r 兲兲
2
dr̃
r
共 r̃⫺r ⬘ 兲
4 ␲ r̃ 2
共 R 共 0 兲 共 r 兲兲 2
共 r̃R 共 0 兲 共 r̃ 兲兲 2 ⫹C,
4␲
冉冕
1
共 R 共 0 兲 共 r 兲兲 2 ⫺
4␲
⫹
冕 冉␦
⬁
2
共 rR
冕
dr
2
共 rR
冕
C⫽
共0兲
共 r 兲兲
2
dr
冕
⬁
r
2
共 rR
共 R 共 0 兲 共 r 兲兲 2 f 共 r 兲 r 2 dr⫽0,
共0兲
共 r 兲兲 2
␪ 共 r ⬘ ⫺r 兲
冊
e ⫺2r ⬘
关 ⫺2⫹4r ⬘ 2 ⫹r ⬘ 共 ⫺14⫹8 ␥ ⫹ln 256兲 ⫹4r ⬘ ln r ⬘ 兴 ,
4␲r⬘
共C3兲
e ⫺2 共 r ⬍ ⫹r ⬎ 兲
␹ 共 r ⬍ ,r ⬎ 兲 ⫽
关 ⫺ 共 r ⬍ ⫹r ⬎ 兲 ⫹r ⬎ e 2r ⬍
␲ 2r ⬍r ⬎
共C1兲
A change in the potential equal to ␩ 关 ␦ (r⫺r ⬘ ) 兴 /4␲ r 2 ,
produces a change in the density having a term linear in ␩
equal to ␩ ␹ (r,r ⬘ ), cf. Eq. 共4兲, where ␩ is infinitesimal. ␹ r is
obtained by constructing the first-order change in the density
due to v (1) (r)⫽ 关 ␦ (r̃⫺r ⬘ ) 兴 /4␲ r 2 . For this potential, E (1) is
given by
共0兲
⬁ ⫺t
where Ei(x)⫽⫺ 兰 ⫺z
e /t dt, and r ⬍ is min(r,r⬘). The constant C is determined by the requirement of orthogonality
between ⌿ (0) and ⌿ (1) :
dr̃ 关v 共 1 兲 共 r̃ 兲 ⫺E 共 1 兲 兴
⫻ 共 u 共 0 兲 共 r̃ 兲兲 2 ⫹C.
共1兲
共C2兲
where ␥ is Euler’s constant ( ␥ ⬇0.577 215 664 9). The firstorder density change and thus ␹ r (r,r ⬘ )) is 2(R (0)
(r)) 2 ( f /4␲ ), using Eqs. 共C2兲 and 共C3兲 one gets
which can be integrated once more to
f 共 r 兲 ⫽⫺
册
yielding
If u (0) is nodeless, a first integration yields as u (0) (r→⬁)
→0:
2
df
⫽⫺ 共 0 兲
dr
共 u 共 r 兲兲 2
冋
1
e ⫺2r ⬘
⫹4re ⫺2r ⬘ ⫺4e ⫺2r ⬘ Ei共 2r ⬍ 兲
⫺2
4␲
r
2
⫹4e ⫺2r ⬘ ln r⫹ min关 1,e 2 共 r⫺r ⬘ 兲 兴 ⫹C,
r
⫽0.
This will be solved by using a method proposed by Dalgarno
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drop the subscript l. Multiplying the equation by ⫺2u (0) one
obtains
6833
dr̃ 共 r̃ 2 R 共 0 兲 共 r̃ 兲兲 2 ⫹C.
⫹r ⬍ r ⬎ 兵 ⫺7⫹4 ␥ ⫹2 共 r ⬍ ⫹r ⬎ 兲
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